Problem 86 - Elementary and Intermediate Algebra

Question

Evaluate each expression. $$ 5 \cdot 3^{3}-4 \cdot 2^{3} $$

Step-by-Step Solution

Answer

103

Step 1: Understand the Expression

The given expression is \(5 \cdot 3^{3} - 4 \cdot 2^{3}\). It involves multiplication and exponentiation. We need to perform operations according to the order of operations: parentheses, exponents, multiplication and division, addition and subtraction (PEMDAS/BODMAS).

Step 2: Calculate Exponents

Evaluate the exponents first. Calculate \(3^{3}\) and \(2^{3}\).\[3^{3} = 3 \cdot 3 \cdot 3 = 27\]\[2^{3} = 2 \cdot 2 \cdot 2 = 8\]

Step 3: Perform Multiplications

Use the results from Step 2 to perform the multiplications.Calculate \(5 \cdot 27\) and \(4 \cdot 8\).\[5 \cdot 27 = 135\]\[4 \cdot 8 = 32\]

Step 4: Subtract the Products

Subtract the second product from the first product.\[135 - 32 = 103\]

Step 5: State the Final Result

The expression evaluates to 103.

Key Concepts

Exponents: Understanding Powers

Exponents are a fundamental concept in mathematics that help us understand repeated multiplication. When you see a number with a small superscript, known as the exponent, it tells you how many times to multiply the number by itself.
For example, in the expression \(3^{3}\), the base is 3, and the exponent is 3. This means you multiply 3 by itself three times, which is calculated as follows:

• First, multiply 3 by 3: \(3 \times 3 = 9\)
• Next, multiply the result by 3 again: \(9 \times 3 = 27\)

Therefore, \(3^{3} = 27\). Similarly, \(2^{3}\) involves multiplying 2 by itself three times:

• \(2 \times 2 = 4\)
• \(4 \times 2 = 8\)

Understanding exponents makes it easier to solve complex equations and is a crucial step in the order of operations.

Multiplication: Calculating Products

Multiplication is the operation of scaling one number by another. In the expression \(5 \cdot 3^{3} - 4 \cdot 2^{3}\), after calculating the exponents, we use multiplication to find the products.
Using the earlier calculated exponents, we multiply:

• 5 by 27, which we get from \(3^{3}\):
  \(5 \times 27 = 135\)
• 4 by 8, from \(2^{3}\):
  \(4 \times 8 = 32\)

Multiplication follows the exponents step in the order of operations (PEMDAS/BODMAS). It allows us to simplify the expression, preparing us for the final operation: subtraction.

Evaluation of Expressions: Solving Step by Step

Evaluating expressions involves applying mathematical operations in a particular order to find a final value. The expression \(5 \cdot 3^{3} - 4 \cdot 2^{3}\) must be solved using the correct order of operations, which is crucial to obtain the right result.
Once you've calculated the exponents and performed the multiplications, as shown in the previous sections, you're ready for the final step: subtraction. From our calculations:

• The first product is 135 (from \(5 \times 27\)).
• The second product is 32 (from \(4 \times 8\)).

Now subtract the second product from the first:

• \(135 - 32 = 103\)

Thus, the evaluated result of the expression is 103. Remember, mastering the order of operations ensures precise evaluations and is vital in algebra and beyond.